行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}{-}1 & {-}2 & 0 & 0 \\ 0 & 2 & {-}1 & {-}2 \\ 1 & 2 & 1 & 2 \\ 2 & {-}1 & 1 & {-}1\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 & 0 \\ 0 & 2 & {-}1 & {-}2 \\ 1 & 2 & 1 & 2 \\ 2 & {-}1 & 1 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [0, -2, 2, -1]

\( \qquad\quad = (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 1 & 2 & 1 \\ 2 & {-}1 & 1\end{array}\,\right| + (-1)^{4-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 0 & 2 & {-}1 \\ 2 & {-}1 & 1\end{array}\,\right| + (-1)^{4-4}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 0 & 2 & {-}1 \\ 1 & 2 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-2}\times(-2)\times(-5) + (-1)^{4-3}\times2\times3 + (-1)^{4-4}\times(-1)\times(-2) \)

\( \qquad\quad = 6 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 1 & 2 & 1 \\ 2 & {-}1 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [0, 1, 1]

\( \qquad\quad = (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-2}\times1\times5 + (-1)^{3-3}\times1\times0 \)

\( \qquad\quad = -5 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-2)\times2 = 5 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 1 & 2\end{array}\,\right| = (-1)\times2 - (-2)\times1 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 0 & 2 & {-}1 \\ 2 & {-}1 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [0, -1, 1]

\( \qquad\quad = (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 0 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-2}\times(-1)\times5 + (-1)^{3-3}\times1\times(-2) \)

\( \qquad\quad = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - (-2)\times2 = 5 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 0 & 2\end{array}\,\right| = (-1)\times2 - (-2)\times0 = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}2 & 0 \\ 0 & 2 & {-}1 \\ 1 & 2 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [0, -1, 1]

\( \qquad\quad = (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 1 & 2\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 0 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-2}\times(-1)\times0 + (-1)^{3-3}\times1\times(-2) \)

\( \qquad\quad = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 1 & 2\end{array}\,\right| = (-1)\times2 - (-2)\times1 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}2 \\ 0 & 2\end{array}\,\right| = (-1)\times2 - (-2)\times0 = -2 \)